A few remarks are in order.
First, we only need to sample the
likelihood function
and do not need explicit access to the density \(\pi _{Y\vert X}\) , which might be
intractable or expensive to evaluate.
We are thus in the setting of approximate Bayesian computation (ABC) methods [1]} or,
more broadly, inference in generative models.
Second, we can compute \(\smash{\widehat{T}}\) 
using convex optimization:
we only
need to estimate
a few components, \(\smash{S^{\mathcal {X}}}\) , of the KR
rearrangement \(S\)  that pushes forward
\(\pi _{Y,X}\)  to a standard normal, using
the constrained maximum likelihood estimator
defined in (REF ).
In particular, the block \(\smash{S^{\mathcal {Y}}}\)  defined in (REF ) is not
needed, while
the components of \(\smash{\widehat{S}^{\mathcal {X}}}\)  can be
computed in parallel using (REF ). Third,
the estimation of \(\smash{S^{\mathcal {X}}}\)  in high dimensions can be regularized using the localization techniques of
Section .
(For the purpose of
posterior sampling, we may only care about the bias-variance tradeoff in the estimation of
\(T\) , and not necessarily that of
\(\smash{S^{\mathcal {X}}}\) .)
Fourth, if we constrain the estimator
\(\smash{\widehat{S}^{\mathcal {X}}}\)  to be linear, then
we recover an EnKF with “perturbed observations,” i.e., the stochastic EnKF [2]}.
